Can AB = BA? Solving for Matrix A and B

[soopo]

Homework Statement

Let $$A, B \epsilon R$$^{n x n}.

Homework Equations

A. Show that if AB = BA, then
(A + B)² = A² + 2AB + B².

B. Give an example of 2 x 2 matrices A and B such that
(A + B)² $$\neq$$ A² + 2AB + B².

The Attempt at a Solution

I have tried to find such a matrices A and B such that the requirements applies. Perhaps, this allows to show the equations.

 

[gabbagabbahey]

For part (a), just expand $(A+B)^2$...what do you get when you do that?

 

[soopo]

For part (a), just expand $(A+B)^2$...what do you get when you do that?

$$A^2 +2AB + B^2$$. Do you suggest that this is enough for A?

I accidentally managed to solve B: an example is A= <1, 0; 1, 1> and B= <1, 1; 0, 1>.

 

[gabbagabbahey]

A^2 +2AB + B^2.

this is only true when AB=BA, since

$$(A+B)^2=(A+B)(A+B)=AA+AB+BA+BB=A^2+AB+BA+B^2$$

...do you follow?

 

[soopo]

this is only true when AB=BA, since

$$(A+B)^2=(A+B)(A+B)=AA+AB+BA+BB=A^2+AB+BA+B^2$$

...do you follow?

Good Point! This must be enough for A. Thanks!